CRITERIA BASED ON WALKS 129

1000 0001 1001 0010 0011 1010 1011 0100 0110 1100 0101 0111 1101 1110 1111

Figure 5.2.5. Model AB.

measure additionally the component A. However, if we would mea- sure three instead of two time-points, we have a chance to observe time series that would speak against model fAB (marked in red)

fAB fB 00,10,00 true false 01,00,01 false true 01,11,01 true false 10,00,01 true false 10,00,10 true false 10,11,01 true false 10,11,10 false true 11,01,00 true false 11,01,11 true false 11,10,00 true false .

The remaining time series would be covered by measuring only P again.

• To distinguish Gasync(fAB) and Gasync(fP) we could measure:

B and Ap: We need to measure at least three time-points and only

the sequence 00 → 10 → 00 could eliminate fP.

5.3. Discussion

In this chapter we presented several ideas how different BNs or different possible behaviors of a single BN can be distinguished based on a subset of components of the BN. In Section 5.1 we presented formalizations of such criteria and we were able to use QBF solvers (e.g. Janota[2017] here) to find solutions in smaller systems. The reformulation of theState-Discrimination- Problem in Proposition 5.1 suggests that there exists no algorithm for this type of problems that will scale well. Nevertheless, we were able to demon- strate the usefulness of this approach on a a cell-fate decision model with 11 components in Section 5.1.3 from Calzone et al. [2010]. This suggests that for many biological models it is still possible to obtain helpful results. Also note that the original cell-fate decision model constructed in Calzone et al. [2010] contained 28 components. To analyze this model the original model was reduced inCalzone et al.[2010] using a network reduction method suggested in Naldi et al. [2011]. This shows that combining clever ways of reducing the Boolean network before using the methods presented here can enable us to tackle bigger systems. In Section 5.1.4we gave another sugges- tion by reformulating the State-Discrimination-Problem into the language of algebraic geometry. We showed that the usage of Gröbner bases con- stitutes another possibility to compute the solutions. In the Boolean case there are relatively efficient computational tools for such tasks Brickenstein and Dreyer [2009]. Therefore, this could be a further promising approach to obtain better running times. Another problem is that the criteria in this chapter can lead to too many components that are necessary to measure. Therefore, it could be useful to relax the criteria in Section 5.1by including probabilistic methods. I.e. allowing that the projected sets A and B\A in theState-Discrimination-Problemintersect to a certain degree.

Also the opposite approach – to analyze what can be said about a hy- pothesis given a fixed set of components we measure – is of interest.

5.3. DISCUSSION 131

This approach we followed in Section5.2. Here, we used finite automata to analyze walks projected on some components to distinguish models. As a showcase we considered four interaction graphs. We computed the skeleton of these interaction graphs (see Section4.3) and gave for a given set of com- ponents corresponding sequences of signs we need to measure to distinguish them. Also here we do not expect the methods presented to scale well with the number of components in BNs due to the use of nondeterministic fi- nite automata. Methods for decreasing the average running time or suitable heuristic approaches should be investigated consequently here as well. Re- stricting the length of potential sequences a priori for example would be an easy approach to tackle bigger systems. Another problem of this approach is that measurements are not sufficient to observe a complete projected walk. There are also several theoretical aspects worth investigating still in the fu- ture. For example it would be interesting to investigate which ASTGs are equivalent with respect to certain output-nodes. Such ideas could be useful with respect to so-called network motifs, where the relation of certain reoc- curring patterns in interaction graphs with the dynamics of corresponding BNs is suggested.

Conclusion

In this thesis, several approaches to link ODE models and Boolean mod- els of gene regulatory networks were considered. This is not solely of theo- retical interest, but also of practical relevance as properties preserved across different classes of models of GRNs are unlikely to be model artifacts. This holds especially for Boolean models due to their simplicity. In light of this reasoning, the comparison between different modeling formalisms of GRNs can serve as an indicator as to the validity of results. At the beginning of this thesis, we set ourselves the task of determining the relations between these different classes of models. We made two contributions in this respect. In Chapter 2 we studied a well-known algorithm that takes a Boolean function f as an input and outputs a family of ODE systems. We saw that the resulting ODE-system has trajectories resembling the discrete trajecto- ries, but that in many cases there is no reasonable way to link the dynamics of the ODE-system with the dynamics of the corresponding Boolean net- work. However, looking at simple characteristics such as steady states or trap spaces, made it possible to prove that these properties are preserved during the conversion.

Unfortunately, this result alone is less than satisfying, given the fact, that a very large amount of analysis and research in Boolean networks is invested into properties of the state transition graphs relying on reachabil- ity properties. Conclusions based on such properties are thus built on a very shaky foundation. Firstly, the interpretation of Boolean states in the continuous setting is not straightforward. And secondly, even in those spe- cific situations where the interpretation of continuous data is simpler, the “trajectories” found in Boolean models do not necessarily correspond to tra- jectories in the continuous models. Thus, we believe the construction of Boolean models should be more thoroughly linked to the behavior of ODEs. In Chapter 4 we presented such an approach. There, we compared spe- cific sets of ODE models with sets of Boolean models. For this purpose we considered the theory of qualitative differential equations. This constitutes a more general perspective on the relation between ODE systems and Boolean networks. Instead of looking at the concentrations of species in regulatory networks, we considered their dynamic trends, i.e. whether their concentra- tions are increasing or decreasing. In this approach, nothing more than the interaction graph of a regulatory network is necessary. To link these trends to Boolean networks, we reformulated a condition for possible sign changes in terms of a Boolean state transition graph. Afterwards, we showed that this Boolean state transition graph can be treated as the asynchronous state transition graph of a Boolean network. Thus, this approach constitutes an

6. CONCLUSION 133

illustration of how Boolean networks can arise from families of ODEs, al- lowing an exact interpretation of states and transitions in the corresponding ODEs.

Also in Chapter4, we were able to use a similar technique to analyze sets of Boolean models. We were able to obtain in this case an analogous Boolean state transition graph, which is essentially the same as the one obtained from sets of continuous models. This result is in and of itself interesting. Furthermore, such a graph can be effectively generated and reduced to the asynchronous state transition graph of a Boolean network. In principle, it can be used to investigate restrictions on the dynamics in certain sets of asynchronous Boolean networks. This is an area ripe for future investigation. The focus of this work lies on the rather theoretical question of how different modeling formalisms of GRNs compare to one another. More gen- erally, the overall goal, this thesis seeks to realize, is a good understanding of different modeling formalisms of GRNs and the ability to combine them efficiently without relying purely on heuristics. Despite this theoretical ap- proach, we gave throughout this thesis ideas for applications of the presented results. We believe these ideas deserve more attention in the future and con- stitute a natural continuation of this research. In our opinion, the present goal should be to exploit the results in this thesis regarding the relation between different modeling frameworks for concrete applications.

The results in Chapter 2 on trap spaces open the door for combining Boolean and continuous models in better ways. Trap spaces could be used to facilitate parameter estimation by reducing the dynamics of an ODE model to trap spaces obtained from a coarser Boolean counterpart. This should be tested systematically for more models.

In Chapter 4, we considered a specific way of abstracting the dynamics of continuous models. However, this constitutes only a special case in the more general theory of qualitative reasoning Kuipers[1994]. In our opinion, it could be interesting to investigate different ways of abstracting ODEs and linking them to Boolean networks or, more generally, discrete multivalued networks.

Finally, the reduction of the Boolean state transition graphs to asynchro- nous Boolean networks considered in Chapter 4 constitutes a link between ODE models, interaction graphs and Boolean networks which deserves more attention. Similar to the results in Chapter 2, this yields the possibility of using Boolean networks to analyze models in different model classes. This was also the motivation to deviate in Chapter5from the original question of this thesis. Motivated by the relationship between Boolean networks, ODE models and interaction graphs, we began in Chapter 5 to explore ways to take advantage of these relationships. It will be interesting to see how such results will unfold in combination with the ideas of Chapter 5in the future.

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